Render entire scene normals
Got it! As I thought, Blender's normal pass contains exactly the right data, but it's not in the right format.  From a video intending to use the normal pass in After Effects' "Normality" plugin, I found that it's easy to convert Blender's colors to the standard colors using the node editor:

(This node setup is for Blender Render, not Cycles. For a Cycles solution see this answer.)